Question

Let the function $f\left(x\right)=\sin x+\cos x$, be defined in $\left[0,2\pi \right]$, then f(x)

• A. increases in $\left(\frac{\pi }{4},\frac{\pi }{2}\right)$
• B. decreases in $\left[\frac{\pi }{4},\frac{5\pi }{4}\right]$
• C. increases in $\left[0,\frac{\pi }{4}\right]\cup \left[\pi ,2\pi \right]$
• D. decreases in $[0,\frac{\pi }{4})\cup (\frac{\pi }{2},2\pi ]$

Answer 1

The correct option is B decreases in $\left[\frac{\pi }{4},\frac{5\pi }{4}\right]$

For increasing /decreasing we need f'(x)

$f^{\prime }\left(x\right)=\cos x-\sin x=\sqrt{2}(\sin (\frac{\pi }{4}-x))$

it is negative in :$[\frac{\pi }{4},\frac{5\pi }{4}]$: f(x) decreases

it is positive in :$[0,\frac{\pi }{4}]\bigcap [\frac{5\pi }{4},2\pi ]$ :f(x) increases